Can you post the entire /var/log/mongodb/mongod.log to www.pastebin.com and link it here.
It usually happens when mongo crashes (or the entire server is hard rebooted), but without seeing all the logs, it's hard to tell.
But even with the logs, we might not find the solution - it depends on how well the logging was around the crash time.

The server ran out of available RAM at least once (2017-06-08T04:24:23.444+0000) - and it also looks like it had an unclean shutdown (crash or hard reboot) at least once (2017-06-08T10:55:16.809+0000)

I know those dates are not today, but if something like that has happened before, then I would guess it could be the cause of today's problems.
To dig much deeper, we would also have to look at the syslog (and probably other logs too) going back the same period - but most setups don't have logging going back that far unless specifically configured.

I see that it says "Data directory /data/db not found", and indeed there is no such directory. However, it was working fine yesterday, so I don't understand what could have happened. Could my data directory have been deleted remotely somehow?